Digital SATMath: Advanced MathMedium
A physicist is studying the trajectory of a particle. The height of the particle, h(t), in meters, after t seconds, is given by the function h(t) = -t^2 + 8t + 9. What is the maximum height reached by the particle?
- A36 meters
- B9 meters
- C25 meters
- D16 meters
Show answer & explanationAnswer & explanation
Correct answer: C. 25 meters
The maximum height is the y-coordinate of the vertex of the parabolic function. For h(t) = at^2 + bt + c, the t-coordinate of the vertex is -b/(2a). Here, t = -8/(2*-1) = 4. Substitute t=4 into h(t) to find the maximum height: h(4) = -(4)^2 + 8(4) + 9 = -16 + 32 + 9 = 25.
Why the other options are wrong
- A. This is an incorrect calculation, likely a miscalculation of the vertex or substitution.
- B. This is the initial height (when t=0), not the maximum height.
- D. This is an incorrect calculation; possibly 8*4 minus 16, but without adding 9.
Maximum/Minimum of a Quadratic Function
For a quadratic function f(x) = ax^2 + bx + c, if a < 0, the parabola opens downwards and has a maximum value at its vertex. If a > 0, the parabola opens upwards and has a minimum value at its vertex.
- The x-coordinate of the vertex is always -b/(2a).
- The maximum or minimum value is the y-coordinate of the vertex, f(-b/(2a)).
- This concept is crucial for optimization problems.
Memory trick: Vertex's 'y' gives the peak, for the function's highest streak!